Contextual Independence |

**Definition.**
*
Given a set of variables C, a context on C is
an assignment of one value to each variable in C. Usually
C is left implicit, and we simply talk about a context. We
would say that C are the variables of the context. Two
contexts are incompatible if there exists a variable that is
assigned different values in the contexts; otherwise they are compatible. We write the empty context as true.
*

**Definition.**
*[5]
Suppose X, Y, Z and C are sets of variables.
X and Y are contextually independent given
Z and context C=c, where c in dom(C), if
*

P(X|Y=yZ_{1}&=zC_{1}&=c) = P(X|Y=yZ_{2}&=zC_{1}&=c)

*We also say that X is contextually independent of Y given
Z and context C=c.
Often we will refer to the simpler case when the
set of variables Z is empty; in this case we say that X and Y are
contextually independent given context C=c.
*

**Example.**
*
Given the belief network and conditional probability table of Figure
*,
*

*E*is contextually independent of*{C,D,Y,Z}*given context*a&b*.*E*is contextually independent of*{C,D,Y,Z}*given*{B}*and context*a*.*E*is not contextually independent of*{C,D,Y,Z}*given*{A,B}*and the empty context*true*.*E*is contextually independent of*{B,D,Y,Z}*given context.`~`

a&c*E*is contextually independent of*{A,B,C,D,Y,Z}*given*B*and context.`~`

a&`~`

c &d

- Where Does Contextual Independence Arise?
- Parent Contexts and Contextual Belief Networks
- Parent Skeletons
- Contextual Factors

David Poole and Nevin Lianwen Zhang,Exploiting Contextual Independence In Probabilistic Inference, Journal of Artificial Intelligence Research, 18, 2003, 263-313.

Contextual Independence |